Math 2.0

Number theory Definition

Regulator of a number field

Builds on Algebraic number fieldDirichlet's unit theoremDeterminantNatural logarithm

Definition

Let KK be an algebraic number field with ring of integers 𝒪K\mathcal{O}_K. Let σ1,…,σr1\sigma_1, \ldots, \sigma_{r_1} be the real embeddings of KK, and let σr1+1,…,σr1+r2\sigma_{r_1 + 1}, \ldots, \sigma_{r_1 + r_2} be one embedding from each pair of complex conjugate non-real embeddings. Put Nj=1N_j = 1 for j≤r1j \le r_1 and Nj=2N_j = 2 for j>r1j > r_1, and let r=r1+r2−1r = r_1 + r_2 - 1. Let u1,…,uru_1, \ldots, u_r be a system of fundamental units, as in Dirichlet's unit theorem. If r≥1r \ge 1, the regulator of KK is RK=∣det⁡(Njlog⁡∣σj(ui)∣)1≤i,j≤r∣.R_K = \abs{\det\big(N_j \log \abs{\sigma_j(u_i)}\big)_{1 \le i, j \le r}}. This is the absolute value of the determinant of an r×rr \times r matrix. The matrix keeps the first rr of the r+1r + 1 columns j=1,…,r1+r2j = 1, \ldots, r_1 + r_2. If r=0r = 0, the regulator is RK=1R_K = 1.

Remarks

The value does not depend on the choices. Each row of the full r×(r+1)r \times (r+1) matrix sums to 00, because ∏j∣σj(u)∣Nj=∣NK/ℚ(u)∣=1\prod_j \abs{\sigma_j(u)}^{N_j} = \abs{N_{K/\Q}(u)} = 1 for a unit uu; so deleting another column changes only the sign of the determinant. Another system of fundamental units differs by a matrix in GLr(ℤ)\mathrm{GL}_r(\Z), which has determinant ±1\pm 1. Here log⁡\log is the natural logarithm.

Example: for K=ℚ(2)K = \Q(\sqrt{2}), r=1r = 1 and u1=1+2u_1 = 1 + \sqrt{2}, so RK=∣log⁡(1+2)∣≈0.881R_K = \abs{\log(1 + \sqrt{2})} \approx 0.881.

Non-example: in ℚ(2)\Q(\sqrt{2}), the unit 3+22=(1+2)23 + 2\sqrt{2} = (1 + \sqrt{2})^2 is not fundamental, since it does not generate the units modulo ±1\pm 1. Using it in the formula gives 2log⁡(1+2)2 \log(1 + \sqrt{2}), which is twice the regulator.

For ℚ\Q and for imaginary quadratic fields, r=0r = 0 and RK=1R_K = 1.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptDeterminant, concept, no evidence levelDeterminantconceptFree abelian group, concept, no evidence levelFree abelian groupconceptGroup of units of a ring, concept, no evidence levelGroup of units of a r…conceptLattice (group), concept, no evidence levelLattice (group)conceptMinkowski's theorem, statement, no evidence levelMinkowski's theoremstatementNatural logarithm, concept, no evidence levelNatural logarithmconceptRegulator of a number field, concept, no evidence levelRegulator of a number…conceptRoot of unity, concept, no evidence levelRoot of unityconceptDirichlet's unit theorem, statement, no evidence levelDirichlet's unit theo…statementRegulator of a number field, definition, no evidence levelRegulator of a number…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Determinant concept
  3. Free abelian group concept
  4. Group of units of a ring concept
  5. Lattice (group) concept
  6. Minkowski's theorem statement
  7. Natural logarithm concept
  8. Regulator of a number field concept
  9. Root of unity concept
  10. Dirichlet's unit theorem statement
  11. Regulator of a number field definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem
About this page
Tag3AG1JV6J80
URIm2://3AG1JV6J80
Kinddefinition
WritingW0 stub
MSC11R27, 11R29
DefinesRegulator of a number field
UsesAlgebraic number field, Dirichlet's unit theorem, Determinant, Natural logarithm
Changed2 November 2026

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